The Irreducible Representations of the Lie algebra su(2) Ivan Vuković Advisor : prof. dr. sc. Saša Krešić-Jurić Split, September 2015 University of Split Faculty of Natural and Mathematical Sciences Department of Physics This page intentionally left blank. 2 Contents 1 Introduction 1 2 Lie groups and Lie algebras 2 3 Representations 6 3.1 Lie algebra su(2) .............................7 3.2 Lie algebra so(3) .............................8 4 SU(2) and SO(3) 9 5 Irreducible representations of su(2) 15 6 Physical significance 19 7 Concluding Remarks 23 i 1 Introduction Symmetry is arguably the most important concept in physics, whether it be rota- tional symmetry, Lorentz symmetry, or some other symmetry. In many important cases, these symmetries form a continuous group, i.e. a group parameterized by real numbers. Mathematically, these groups are often Lie groups, smooth manifolds which are also groups. The tangent space at the identity element of the Lie group is a vector space that has a natural bracket operation that makes it a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group and its linearity makes it easy to work with. In quantum mechanics, symmetry is encoded through a unitary action of the group on the relevant Hilbert space. By Noether’s theorem, to every differentiable sym- metry generated by local actions (i.e. actions of a Lie group), there corresponds a conserved physical quantity. Observables corresponding to conserved quantities form a representation of the Lie algebra of that group. For example, invariance of the Hamiltonian operator Hˆ of a quantum system under rotations means that Hˆ commutes with the relevant representation of the rotation group and thus also with the associated Lie algebra operators. This commutativity, in turn, implies that the eigenspaces for Hˆ are invariant under rotations. At this point, the importance of Lie algebras and their representations should be evident. In this paper, we investigate the special unitary group SU(2) and the rotation group SO(3). The group SU(2) is particularly important as a double cover of SO(3). We investigate the physical meaning of SO(3) by exploring the Lie algebra su(2) of its double cover, SU(2). In particular, we will compute all finite-dimensional irreducible representations of su(2) up to isomorphism. The close relationship of so(3) with su(2) will prove to be especially illuminating. Basic knowledge of group theory and analytical mechanics is assumed throughout the text. 1 2 Lie groups and Lie algebras To prepare the ground, first we have to understand what Lie algebras and their representations are. Definition 1. A Lie algebra over a field F (char F 6= 2) is a vector space g over F, together with a map (called the Lie bracket) [ · , · ]: g × g −→ g with the following properties: (∀a, b ∈ F), (∀X, Y, Z ∈ g) 1. Bilinearity: [aX + bY, Z] = a[X, Z] + b[Y, Z] [Z, aX + bY ] = a[Z, X] + b[Z, Y ] 2. Anticommutativity: [X, Y ] = −[Y, X] 3. The Jacobi identity: h i h i h i X, [Y, Z] + Y, [Z, X] + Z, [X, Y ] = 0 If F = R (F = C), we say that g is a real (complex) Lie algebra. If g is a finite-dimensional space, we say that g is a finite-dimensional Lie algebra. A large class of Lie algebras may be obtained by the following procedure. Proposition 1. Let A be an associative algebra and let g be a subspace of A with the property that for all X, Y ∈ g, XY − YX is again in g. Then the bracket [X, Y ] := XY − YX makes g into a Lie algebra. This bracket is called the commutator of X and Y . Proof. The proof trivially follows from the definition of the Lie algebra. So far, we have discussed Lie algebras as abstract algebraic structures. Let us now turn to Lie algebras of Lie groups. First we define matrix Lie groups. 2 Definition 2. Let G be a subgroup of GL(n; C). Then, G is called a matrix Lie group if for every sequence of matrices in G: (Am −→ A) =⇒ A ∈ G or A/∈ GL(n; C) In other words, a matrix Lie group is a (topologically) closed subgroup of GL(n; C). Let us list the matrix Lie groups that will be used in this paper. They are also some of the most important examples of Lie groups in general. Examples of matrix Lie groups • GL(n; R) - invertible real matrices • GL(n; C) - invertible complex matrices • SL(n; R) - invertible real matrices with unit determinant • SL(n; C) - invertible complex matrices with unit determinant • O(n) - orthogonal real matrices • SO(n) - orthogonal real matrices with unit determinant • U(n) - unitary matrices • SU(n) - unitary matrices with unit determinant • Sp(n) = (Sp(n; C) ∩ U(2n)) ⊂ Mn(H) - compact symplectic group (quaternionic unitary matrices) To associate a Lie algebra to a matrix Lie group, we need the notion of the expo- nential of a matrix. X Definition 3. Let X ∈ Mn(C). The matrix exponential e ≡ exp(X) is defined as exp : Mn(C) → GL(n; C) ∞ Xm eX = X . m=0 m! The matrix exponential shares some but not all of the properties of the exponential of a number. We list some of these properties without proof. 3 Properties of the matrix exponential • eX is a continuous function of X ∞ P Xm • the series m! converges for all X ∈ Mn(C) m=0 • e0 = I • det(eX ) 6= 0 • eXT = (eX )T • eX† = (eX )† • (eX )−1 = e−X • e(α+β)X = eαX eβX (∀α, β ∈ C) • XY = YX =⇒ eX+Y = eX eY = eY eX −1 • C ∈ GL(n; C) =⇒ eCXC = CeX C−1 d tX • dt e = X t=0 X+Y X Y m • e = lim (e m e m ) (Lie product formula) m→∞ • det(eX ) = etr(X) Definition 4. Let G be a matrix Lie group. The Lie algebra of the matrix Lie group G, denoted g is defined as: n o tX g = X ∈ Mn(C) e ∈ G, ∀t ∈ R , where etX is the matrix exponential of tX. In Lie algebras of matrix Lie groups, the matrix commutator takes the role of the Lie bracket. Definition 5. Let A, B ∈ Mn(C). The commutator of A and B is given by [A, B] = AB − BA. Intuitively, if we think of Lie groups as continuous groups, elements of its Lie algebra can be interpreted as differential group elements, i.e. group infinitesimals. 4 In the general theory of Lie groups, the Lie algebra of a Lie group is geometrically thought of as the tangent space to the identity element and the exponential map is defined as follows. Definition 6. Let G be a matrix Lie group and g its Lie algebra. The expo- nential map is a map exp : g −→ G, where exp(X) ≡ eX is the matrix exponential given by Definition 3. Every Lie group homomorphism gives rise to a Lie algebra homomorphism in a natural way. Definition 7. Let g and h be Lie algebras over a field F. A linear map ϕ : g → h such that ϕ([X, Y ]) = [ϕ(X), ϕ(Y )] (∀X, Y ∈ g) is called a Lie algebra homomorphism. If ϕ is a bijective map, we say that ϕ is a Lie algebra isomorphism. Theorem 1. Let G and H be matrix Lie groups with Lie algebras g and h, respectively. Suppose Φ: G → H is a Lie group homomorphism. Then there exists a unique linear map ϕ : g → h such that Φ(etX ) = etϕ(X) for all t ∈ R and X ∈ g. This linear map has the following additional proper- ties: (∀X, Y ∈ g), (∀A ∈ G) • ϕ ([X, Y ]) = [ϕ(X), ϕ(Y )] • ϕ(AXA−1) = Φ(A)ϕ(X)Φ(A)−1 • ϕ(X) = dΦ dt t=0 An important fact follows from the preceding theorem. G ∼= H =⇒ g ∼= h Furthermore, every matrix Lie group is a real embedded submanifold of Mn(C) whose dimension is equal to the dimension of its Lie algebra as a real vector space. 5 3 Representations Definition 8. Let G be a matrix Lie group. A finite-dimensional complex representation of G is a Lie group ho- momorphism Π: G −→ GL(V), where V is a finite-dimensional complex vector space. If G is a real matrix Lie group, then a finite-dimensional real represen- tation of G is a Lie group homomorphism where V is a finite-dimensional real vector space. Let g be a real (or complex) Lie algebra. A finite-dimensional complex representation of g is a Lie algebra homomorphism π : g −→ gl(V), where V is a finite-dimensional complex vector space. If g is a real Lie algebra, then a finite-dimensional real representation of g is a Lie algebra homomorphism where V is a finite-dimensional real vector space. If Π or π is an monomorphism, the respective representations are called faith- ful. It is fruitful to think of representations as actions of groups or Lie algebras on a vector space. Definition 9. Let G be a group and let X be a set. An action of the group G on X is a map · : G × X → X, denoted (g, x) 7→ g · x, satisfying (∀x ∈ X), (∀g, h ∈ G) 1. e · x = x 2. g · (h · x) = (gh) · x, where e is the identity element in G. 6 A representation Π of G on some vector space V gives rise to a linear action of G on V , given by: g · v := Π(g)v Thus, for convenience and clarity, we may use g · v as an alternative notation to Π(g)v.